Do the elder and younger brothers have the right to claim their share in the inheritance of the rented apartment that their mother used to live in and passed away in, knowing that the middle brother is the one who lives in it and has been paying its rent for a long time, and how is the inheritance calculated if they have a legitimate right?

The rented apartment is not the mother's property; rather, it belongs to the state or its private owner. Therefore, the heirs have no right to it. The aforementioned lease contract is an invalid contract that is not inherited from the mother. It is not permissible for your brother to continue with this contract, which inflicts injustice upon the owner and is considered consuming his wealth unjustly. A Muslim must reject laws that contradict Allah's ruling.